Method and system for identifying the onset of a turbulent boundary layer induced by a body moving through a fluid medium

ABSTRACT

A method of detect the onset of turbulence in connection with a body movinghrough a fluid medium. First, the body is supplied with sensors each for generating a signal suitable for measuring amplitude of pressure fluctuations of the medium proximate a region of said sidewall of the body in at least a region of the body in which turbulence is expected to occur. During a reference stage during which the body moves through the fluid medium when it is known that turbulence is occurring around at least a portion of said body , the sensors each generate reference temporal pressure data representing fluctuations in pressure of the fluid medium around said body. In response to reference temporal pressure data generated by sensors in a turbulence zone at which turbulence is occurring and sensors in a transition zone between the turbulence zone and a laminar flow zone, a method-of-delay phase portrait is generated for each of a progression of selected delay intervals. These operations are repeated during an operational stage, and phase portraits generated during the operational stage are compared to phase portraits in response to the reference temporal pressure data from the transitional zone and the turbulence zone, for corresponding ones of said selected delay intervals,, and a determination of the onset of turbulence is made in response to such comparison.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein may be manufactured by or for theGovernment of the United States of America for Governmental purposeswithout the payment of any royalties thereon or therefor.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

The instant applications is related to two-co-pending U.S. PatentApplications entitled METHOD AND SYSTEM FOR SUPPRESSING NOISE INDUCED INA FLUID MEDIUM BY A BODY MOVING THERETHROUGH U.S. patent applicationSer. No. 117,513 filed on Sep. 2, 1993 (Navy Case No. 75551; and METHODAND SYSTEM FOR REDUCING DRAG ON A BODY MOVING THROUGH A FLUID MEDIUMU.S. patent application Ser. No. 117,512 filed on Sep. 2, 1993 (NavyCase No. 75552), both by the same inventor and filed on the same date asthis patent application.

BACKGROUND OF THE INVENTION

(1) Field of the Invention

The invention is generally related to the field of signal processing,and more specifically to the determination of the onset of turbulence inconnection with a body moving through a fluid medium.

(2) Description of the Prior Art

The boundary layer flow over a body which has an axisymmetric exteriorsurface moving through a stationary fluid, in which the motion is steadyand directed parallel to the longitudinal axis of the body, may becharacterized by three spatially separated, but somewhat over-lapping,flow zones. These zones may be described as a laminar zone, atransitional zone and a turbulent zone. In the laminar zone, which isgenerally located toward the leading edge of the body, there are noappreciable disturbances of the pressure of the fluid on the surface ofthe body, and hence any measured pressure fluctuations within this zoneare appreciably negligible. As the flow develops downstream of theleading edge of the body, it enters into the transition zone. Thetransition zone evolves from the latter stages of the laminar flow,where infinitesimal, linear wavelike disturbances, so-calledTollmien-Schlichting (T-S) waves, develop and begin to amplify bothtemporally and spatially with distance downstream of the leading edge ofthe body. The position along the wall at which small disturbance wavesbegin to amplify is related to both the shape and size of the body, aswell as inertial characteristics of the flow of the medium in thevicinity of the wall. A re9ion of decreasing velocity (or increasingpressure) of the fluid relative to the surface of the body downstream ofthe leading edge of the body marks the beginning of a zone where anadverse pressure gradient sets in. This adverse pressure gradient has adestabilizing effect. At or shortly downstream of this position, T-Sdisturbance waves would be expected to start to grow.

The amplitude of the T-S waves becomes larger as they convectdownstream, and as a result their evolution becomes nonlinear andturbulent bursting is observed. The bursts initially are local and occurintermittently over each point over this portion of the surface of thebody. The number of bursts per unit time increases with distance alongthe surface from the leading edge of the body. Downstream, the burstingfinally coalesces in such a way that the flow reaches a fully turbulentstate. The position along the wall where bursting fully coalesces is thedividing line between the end of the transitional zone and the start ofthe turbulent zone.

Nonlinear coupling of energetic modes in the spectra of fluctuations inthe velocity of the medium proximate the sidewall of the body, or offluctuations in the pressure exerted by the medium on the sidewall ofthe body following T-S wave amplification. The nonlinear nature of thetransition produces a temporal power spectrum of frequencies of the T-Swaves and combinations of the sums and differences of the respectivefrequencies. In the time domain, the time scales of interest are thereciprocals of the associated T-S frequencies. These principal timescales are characterized by a value corresponding to the wavelengths ofthe T-S waves divided by their convective velocities.

SUMMARY OF THE INVENTION

The invention provides a new and improved method of predicting thedynamics of a wall-bounded shear flow of a fluid along an axisymmetricbody exhibiting steady motion through the fluid.

In brief summary, the method in accordance with the invention detectsthe onset of turbulence in connection with a body moving through a fluidmedium. First, the body is supplied with sensors each for generating asignal suitable for measuring amplitude of pressure fluctuations of themedium proximate a region of said sidewall of the body in at least aregion of the body in which turbulence is expected to occur. During areference stage during which the body moves through the fluid mediumwhen it is known that turbulence is occurring around at least a portionof said body , the sensors each generate reference temporal pressuredata representing fluctuations in pressure of the fluid medium aroundsaid body. In response to reference temporal pressure data generated bysensors in a turbulence zone at which turbulence is occurring andsensors in a transition zone between the turbulence zone and a laminarflow zone, a method-of-delay phase portrait is generated for each of aprogression of selected delay intervals. During an operational stage,during which the body moves through the fluid medium, the sensors eachgenerate operational temporal pressure data representing fluctuations inpressure of the fluid medium around said body. In response to theoperational temporal pressure data, a method-of-delay phase portrait isgenerated for each of a progression of selected delay intervals. Thephase portraits generated in response to the operational temporalpressure data during the operational stage are compared to phaseportraits generated in response to the reference temporal pressure datafrom the transitional zone and the turbulence zone, for correspondingones of said selected delay intervals. This is done to determine whetherthe phase portraits generated in response to the operational temporalpressure data have selected characteristics present in the phaseportraits generated in response to the reference temporal pressure datafrom the transitional zone and the turbulence zone. In turn, adetermination of the onset of turbulence is made in response to suchcomparison.

In another aspect, the invention provides a system for detecting theonset of turbulence in connection with a body moving through a fluidmedium. The body is supplied with sensors each for generating a signalsuitable for measuring amplitude of pressure fluctuations of the mediumproximate a region of said sidewall of the body in at least a region ofthe body in which turbulence is expected to occur. During a referencestage during which the body moves through the fluid medium when it isknown that turbulence is occurring around at least a portion of saidbody , the sensors each generate reference temporal pressure datarepresenting fluctuations in pressure of the fluid medium around saidbody. In response to reference temporal pressure data generated bysensors in a turbulence zone at which turbulence is occurring andsensors in a transition zone between the turbulence zone and a laminarflow zone, means generate a method-of-delay phase portrait for each of aprogression of selected delay intervals. During an operational stage,during which the body moves through the fluid medium, the sensors eachgenerate operational temporal pressure data representing fluctuations inpressure of the fluid medium around said body. In response to theoperational temporal pressure data, means generate a method-of-delayphase portrait for each of a progression of selected delay intervals.Comparators compare the phase portraits generated in response to theoperational temporal pressure data during the operational stage to phaseportraits generated in response to the reference temporal pressure datafrom the transitional zone and the turbulence zone, for correspondingones of said selected delay intervals, to determine whether the phaseportraits generated in response to the operational temporal pressuredata have selected characteristics present in the phase portraitsgenerated in response to the reference temporal pressure data from thetransitional zone and the turbulence zone, with the comparator furthermaking a determination of the onset of turbulence is made in response tosuch comparison.

In one embodiment, the reference temporal pressure data and operationaltemporal pressure data are low-pass filtered prior to generation of thephase portraits. In that operation, the data generated by each sensorover a selected time window is Fouriertransformed to 9enerate powerspectra, and an inverse-Fourier transform is applied to the Fouriercoefficients associated with

frequencies below a selected cut-off frequency, the result being used ingenerating the respective phase portraits. In addition, further indiciaof the turbulence state of the medium is generated from the powerspectra generated during both the reference stage and the operationalstage.

BRIEF DESCRIPTION OF THE DRAWINGS

This invention is pointed out with particularity in the appended claims.The above and further advantages of this invention may be betterunderstood by referring to the following description taken inconjunction with the accompanying drawings, in which:

FIG. IA is a diagram of an axisymmetric body with which the inventivemethod is used, and FIG. lB is a diagram of a pressure sensor mounted onthe body which is used to gather data used in practicing the method;

FIG. 2 is a flow chart illustrating the new method in accordance withthe invention;

FIGS. 3A through 3F are a series of curves depicting pressure spectravalues as a function of sensor location in a series of sensorssuccessively located in the downstream direction along the axisymmetricbody of FIG. 1;

FIG. 4 depicts a value which is a measure of overall power (P_(rms)) asa function of such series of sensors; and

FIGS. 5A dna 5B respectively are two different families of phaseportraits generated using signals from one of such series of sensors,with the individual portraits of each family representing a series ofphase delay itnervals, the family of portraits of FIG. 5A being with thesensor in a transition zone and the family of portraits of FIG. 5B beingwith the sensor in a turbulence zone.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Prior to describing the new method, it would be helpful to describe anaxisymmetric body with which the invention is used. FIG. 1A is a diagramof such a body 10, in particular representing the leading edge 11 of thebody 10 and a portion of the body 10 downstream of the leading edge. Thebody 10 moves through a fluid medium (not shown) along a directionparallel to an axis 12, around which the body 10 is symmetric. As shownin FIG. 1A, flow of the medium around the body is generally laminar in azone proximate the leading edge 11. Downstream of the laminar zone is atransition zone where Tollmein-Schichting (T-S) waves, which begin nearthe border between the laminar zone and the transition zone, tend togrow in amplitude according to a growth characteristic which istypically exponential. The waves represent fluctuations in pressure ofthe medium against the sidewall of the body 10 at points in thetransition zone. In the transition zone, the waves are generally welldefined combinations of particular frequencies which may depend on anumber of variables, including the viscosity of the fluid forming themedium, the geometry (the particular size and shape) of the body 10, andthe speed of the body 10 through the medium. The waves become fullyturbulent, exhibiting a broadband spectrum of frequencies in thepressure fluctuations around the body, in a turbulence zone locateddownstream of the transition zone.

In practicing the method, pressure sensors are mounted in the sidewallof the body 10, one of which, namely, sensor 20, is shown mounted in thesidewall in FIG. IB. In one embodiment, the pressure sensor 20 is of thepiezoelectric type having a cylindrical shape. The senso 20 is insertedin and bonded to an opening in the sidewall 21. The exterior end of theopening through sidewall 21 is provided with an inwardly taperedstainless steel grommet 21a which is in direct contact with sensor 20.An insert sleeve 21b of a suitable packing material is bonded within theopening through sidewall 21 and extends from the exterior end of grommet21 to the interior end of the opening through sidewall 21. Anelastomeric covering 22 extends over sidewall 22 and over the end faceof pressure sensor 20. A microdot connector 23 on the pressure sensor 20facilitates connection of the pressure sensor 20 to data processingequipment 24 which practices the method. In one embodiment, the sensorsare mounted in rows along the length of the body 10, the rows beingparallel to the axis 12 and separated around the circumference of thebody 10 by approximately ninety degrees. The pressure sensors 20 in eachrow are separated by a uniform separation, approximately two inches inone embodiment. As between rows, the sensors 20 are staggered topreclude mutual interference between sensors, yet provide minimalseparation alon9 the axis 12, so that a full set of measurements can beobtained from the early transition zone (that is, the portion of thetransition zon proximate the laminar zone) fully into the turbulencezone. In one embodiment, the rows are staggered so that the positions ofthe pressure sensors 20 in rows on opposite sides of the body 10 aredisplaced by approximately one-half inch, and pressure sensors inproximate rows along body 10 are displaced by approximately one-quarterinch. 4. With this background, the method for detecting the onset ofturbulence will be described in connection with FIGS. 2 through 5B. Withreference initially to FIG. 2, the method is generally performed in twostages, namely, a reference stage, comprising steps 100 through 102, andan operational stage, comprising steps 103 through 106. In the referencestage, reference data is collected using the body 10 when turbulence isknown to be taking place, and processed to produce graphical displays ornumerically computed values of the processed data taken from thetransitional zone and from the turbulence zone. In this way, differencesbetween displays of processed data taken from the transitional zone andthose taken from the turbulence zone can be identified. In addition,characteristics of the displays of processed data taken from theturbulence zone can also be identified. In the operational stage,operational data is collected, processed and displayed in a mannersimilar to the reference stage, and a determination is made as towhether turbulence is developing along the body 10 based on a visualcomparison between corresponding displays of the processed data obtainedduring the operational stage and the reference stage, or alternativelythe comparison may be numerically computed. The operations performedduring the operational stage may be performed continually, tocontinually provide updated information as to the state of turbulencearound the body 10.

More specifically, during the reference stage, the body 10 is movedthrough the fluid medium in a direction parallel to its axis 12. Duringoperations performed in the reference stage, movement of the body 10 issuch that a turbulence zone is developed at some point along the body10, so that all three zones described above, including the laminar zone,the transition zone and the turbulence zone are present along the body10 from the leading edge 11 in the downstream direction along the axis12. During such movement, fluctuations in the pressure of the fluidagainst the sidewall of the body 10 is sensed by the various sensors 20positioned therealong, and the amplitude of the pressure fluctuations isrecorded digitally at successive points in time to provide pressureamplitude data values p(t) (step 100). The pressure amplitude datavalues p(t) recorded during step 100 are processed to generate both atemporal power spectrum (step 101) and phase portraits (step 102) foreach sensor. The temporal power spectrum indicates the amplitude ofvarious frequencies in the variations and fluctuations in pressure assensed by the particular sensor 20. The phase portraits, generally,provide a visual display or numerically computed indication of thedegree of correlation of phases of the variations and fluctuations inpressure as sensed by the sensor 20, and thus provides a visualindication of whether a turbulence zone is present along the body 10.

In particular, in step 101, a power spectrum is generated for eachsensor. The power spectrum corresponds to the Fourier transform of thepressure amplitude data values, and, for each sequence p(t_(n)) (n=1, 2,. . . N) of pressure amplitude data values for "N" successive points intime, the Fourier transform is

    P(σ.sub.m)=(1/N) Σp(t.sub.n) exp(2iπnm/n)   (Eqn. 1)

(sum taken over "n" from 1 to N, that is, for each pressure amplitudedata value in the sequence) where each P(σ_(m)) represents a Fouriercoefficient, and

    σ.sub.m =m(2π/T),                                 (Eqn. 2)

where T it the total time along which the sequence p(t_(n)) was taken.In one particular embodiment, the Fourier transform is performed using aconventional fast Fourier transform (FFT) operation. The spectrum valuesare normalized, and the result is windowed and smoothed in aconventional manner.

Various spectra for pressure amplitude data values collected fromvarious sensors 20 in the transitional and turbulence zones in oneembodiment are generally depicted in FIGS. 3A through 3F. Each of FIGS.3A through 3F depicts pressure spectra based on data taken from sensors20 ranging from those near the leading edge 11 progressively downstream,with the FIG. 3A depicting pressure spectrum based on data taken from asensor 20 relatively near the leading edge 11 and each successively oneof FIGS. 3B through 3F depicting pressure spectra based on data takenfrom a sensors 20 in successively downstream locations. The graph inFIG. 3A depicts pressure spectra based on data taken from a sensor inthe transitional zone, and the graph in FIG. 3F depicts pressure spectrabased on data taken from a sensor well into the turbulence zone. Theabscissa of each graph represents frequency, with frequency increasingfrom left to right, and the ordinate of each graph represents a functionof the normalized, windowed and smoothed Fourier coefficients (bothabscissa and ordinate are represented in arbitrary units).

Several points are evident from an examination of the series of graphsdepicted in FIG. 3. First, the increase in the peak values from thetop-most graph in FIG. 3A to the second graph in FIG. 3B, both of whichare based on data provided by sensors 20 in the transitional zone,corresponds to observed increases in overall root-mean-square values forthe spectra. The root-mean-square values, which are shown graphically inFIG. 4, are derived from the Fourier coefficients according to therelation

    P.sub.nms =sqrt(|P(ρ.sub.m)|.sup.2)  (Eqn. 3)

where |P(ρ_(m))|² is the dot product of the sequence of Fouriercoefficients, considered as a vector, with itself, and "sqrt" is thesquare-root function. The root-mean-square value associated with thedata provided by each sensor 20 provides a measure of the overall powercontent of the pressure fluctuations as sensed bY the sensor 20. Asshown in FIC. 4, the root-mean-square values increase markedlY in theupstream portion of the transition zone, then decrease toward theturbulence zone, and increase again in the turbulence zone, although theincrease in root-mean-square values in the turbulence zone is much moregradual than in the transition zone. The top two graphs in FIG. 3 in oneembodiment are taken from sensors 20 in the portion of the transitionzone for body 10 in which the root-mean-square values shown in FIG. 4are increasing, and the increase in the peak values from the top graphto the second graph shown in FIG. 3 conforms to the increase inroot-mean-square values shown in FIG. 4 for the associated sensors.

As a second point, broadband spectra such as are shown in theprogression evident from the graphs in FIGS. 3A and 3B are common fornatural transition processes even when the transition is in an earlystage. This follows from two factors. First, general, broadbandbackground disturbances are being amplified as a result of the movementof the body 10 through the fluid medium, so that lower powered pressurewaves or fluctuations detected by the sensors 20 located progressivelydownstream but disposed toward the leading edge 11 of the body 10 are inthe transition zone. The second factor is that there is little or nophase coherence in the pressure waves or fluctuations, resulting in abroadening of the frequency content of the pressure waves orfluctuations as detected by sensors located progressively downstreamover the body 10.

A third observation from the power spectra curves shown in FIGS. 3Athrough 3F is that both the low- and high-frequencies fill in asdetected by the sensors 12 located progressively downstream over thebody 10. This observation follows from the fact that FIGS. 3A and 3B,for example, representing the power spectra as detected by sensorslocated toward the leading edge 11 of body 10 in the early transitionzone, the curves have several well-defined peaks, showing that thefrequency content of the waves detected by those sensors are generallywell defined by the peaks. The peaks generally smooth out progressivelyfrom the curves in FIGS. 3A and 3B toward the curves in FIGS. 3E and 3F,which represent power spectra as detected by sensors 20 in theturbulence zone. The curves in FIGS. 3E and 3F, representing the spectradetected in the turbulence zone, is characterized by a broadbandcontinuum of frequencies, of generally uniform amplitude distribution,over a wide band of frequencies.

As the Fourier coefficients which represent the power spectra aredeveloped in step 101 for the signal generated by each sensor 20,"m"-dimensional phase portraits are also generated for each signal fromthe time series representation of each signal using a conventional"method-of-delays" technique (step 102). In conjunction with thatoperation, in one embodiment the signal is low-pass filtered and, foreach time "t_(k) ", a point in the phase portrait is generated definedwith the coordinates:

    [L(t.sub.k), L(t.sub.k +T), L(t.sub.k +2T), L(t.sub.k +3T), . . . L(t.sub.k +(m-1)T)]                                                 (Eqn. 4)

where "m" is the number of dimensions of the phase portrait, "L(t)" isthe amplitude of the signal received by the sensor 20 at time "t", and"T" is an arbitrarily-selected time delay (hence the name of thetechnique as the "method-of-delays" technique). In one particularembodiment, in which the body 10 is moving through a water medium, delayintervals of less than the period of the Tollmein-Schichting frequencywas used; in particular, "T" is generally selected as a percentage ofthe zero-crossing of the auto-correlation function of the signals or theminimum value of the mutual information function which is consistentwith the selected values.

In one particular embodiment, prior to actual generation of the actualphase portraits, the signals from the sensors 20 are low-pass filtered.In this operation, the results generated in step 101, namely, theFourier coefficients, may be used as an input to step 102, instead ofusing the actual digitized temporal signal data obtained from thesensors 20 in step 100. In that operation, an inverse-Fourier transformoperation is performed using only the Fourier coefficients associatedwith frequencies below a selected cut-off frequency to regeneratedigitized temporal signal data. By ignoring the coefficients associatedwith frequencies above the selected cut-off frequency, those frequenciesare filtered out of the temporal signal data, effectively providinglow-pass filtered temporal signal data.

A phase portrait associated with a signal from a sensor 20 effectivelyrepresents, the time evolution of the signal. FIGS. 5A and 5B depicttwo-dimensional phase portraits generated from one embodiment of theinvention, FIG. 5A being associated with a signal from a sensor 20 inthe transition zone, and FIG. 5B being associated with a signal from asensor 20 in the turbulence zone. Each figure includes six phaseportraits, each of an incrementally lon9er delay parameter, that is ofincrementally-longer time lag. Thus, FIG. 5A includes phase portraits50(1) through 50(6), which are symbolically identified 4. in this textby reference numeral 50(i), with each phase portrait 50(i) representinga delay parameter, or time lag, "T" (from equation Eqn. 4) correspondingto a time interval proportional to the index "i." Similarly, FIG. 5Bincludes phase portraits 51(1) through 51(6), which are in a similarmanner symbolically identified by reference numeral 5i(i). In both FIGS.5A and 5B, in the phase portraits whose index "i" equals 1 [that is, inboth phase portraits 50(1) and 51(1)] the delay parameter "T" usedcorresponds to one sampling period used by the sensor 20 to generate thepressure amplitude data values p(t) used in generating the phaseportrait. In one embodiment "T" corresponds to a time period of on theorder of a tenth of a millisecond.

Each phase portrait geometrically portrays the evolutionary dynamics ofthe associated signal without any temporal averaging or processing(except for that which may be performed by low-pass filtering of thesignals) and showing non-linear coupling of the dynamics and inherentphase relationships up to the dimensional order of the embedded signal.FIG. 5A depicts six (6) phase portraits 50(1), 50(2), 50(3), 50(4) and50(5) and 50(6). The phase portrait 50(1) exhibits a high correlation ofmeasured signal levels at a delay interval corresponding to the samplingperiod of the sensor. It will be appreciated that phase portrait 50(1)is generally ellipsoidal in shape, with the major axis of the ellipselying at approximately a forty-five degree angle to the horizontal[L(t)] axis, and the minor axis being orthogonal to the major axis.Phase portrait 50(2), which is generated using a delay interval twicethat used in phase portrait 50(1), is expanded somewhat along the minoraxis (becoming more circular in shape, with a reduction in the ratio ofmajor axis length to minor axis length) from the phase portrait 50(1).The next phase portrait 50(3) was generated using a delay intervalthrice that used in phase portrait 50(1), and it shows a furtherexpansion along the direction of the minor axis of phase portrait 50(1).Indeed, phase portrait 50(3) becomes an ellipsoid whose major axiscorresponds to the minor axis of phase portrait 50(1).

Phase portraits 50(4) and 50(5), generated using step-wise progressivelylonger delay intervals, show progressive flattening of the ellipses,with increases of the ratios of major axis lengths to minor axislengths. Phase portrait 50(6) generated using a step-wise longer delayinterval than that used for phase portrait 50(5), tends to show abroadening of the minor axis from the phase portrait 50(5). Phaseportraits (not shown) generated using successive step wise longer delayswould be expected to show a further expansion along the minor axes, witheventual return to the orientation of phase portrait 50(1). It should benoted that the ellipsoidal shape is generally reflective of the factthat the time delay "T" is a multiple of one-half wavelength of the T-Swaves in early transition, and the well-formed ellipsoidal shapesgenerally reflect the fact that the pressure fluctuations detected bythe sensor 20 are generally of well defined frequencies, which conformsto the shapes of the power spectra graphs in the transition zone asshown in, for example, FIGS. 3A and 3B.

The phase portraits 51(1) through 51(6) in FIG. 5B were generated usinga signal from a sensor 20 in the turbulence zone at the same delayintervals of the correspondingly-indexed phase portraits 50(1) through50(6) in FIG. 5A. It should be noted that the generally-ellipsoidalshapes of the phase portrait 51(1) through 51(6) generally suggests apersistence of the T-S wave in the turbulence region which is not asreadily apparent from the power spectra shown in, for example, FIGS. 3Eand 3F.

After obtaining the phase portraits (FIGS. 5A and 5B) in step 102 duringthe reference stage, the body 10 may be used in an operational stage,and the results generated during the reference stage may be used todetect the onset of turbulence about the body 10. With further referenceto FIG. 2, in the operation mode, the sensors 20 generate operationalsets of temporal pressure data in the same way as during the referencestage (step 103). The operational temporal pressure data is filtered(step 104) in the same way as in step 101, and method-of-delay phaseportraits are generated (step 105), again in the same way as during thereference stage (specifically step 102).

The phase portraits generated during the operational stage are examinedand compared to corresponding phase portraits generated during thereference stage to determine whether turbulence has been established inthe fluid medium around the body 10 (step 106). In this operation,visual comparisons may be made of phase portraits based on theoperational temporal pressure data with phase portraits of correspondingdelay intervals generated during the reference stage and likely presenceof turbulence detected from similarities in shape of phase portraits ofcorresponding time delays, or alternatively comparisons may benumerically computed. Stated another way, what criterion is being usedfor visual or numerical detection of onset of turbulence is the presenceof charaoteristics of the method-of-delay phase portraits associatedwith presence of turbulence discussed hereinbefore. The comparison maybe performed visually, or alternatively pattern matching apparatus, suchas a suitably trained neural network, may be used in making thecomparison.

The operations described in connection with steps 103 through 106 may beperformed iteratively, to facilitate the detection of onset ofturbulence.

An expanded description of the theoretical basis of the foregoing may befound in Ricard A. Katz, "Transitions to Turbulence: Determinism inNature", a dissertation in the Division of Applied Mathematics, BrownUniversity, Providence, R. I., which was published on May 15, 1993 andwhich is hereby incorporated by reference.

The new method provides a number of advantages. In particular, itprovides a relatively inexpensive and very reliable method for detectingthe onset of turbulence in a fluid medium caused by a body moving in themedium.

It will be appreciated that the reliability of the method 4. may besomewhat limited by noise which may be produced by, for example, sourcesof vibration internal to the body, such as engines which may be used topower the body 10 through the fluid medium. It may be advantageous,prior to beginning the use of the method, to obtain a power spectrum ofsuch sources at each of the sensors 20, which may be used to process thedata obtained from the sensors after the data is acquired, but prior toperforming steps 101 and 102 in the reference stage or steps 104 through106 in the operational stage.

It will be appreciated by those skilled in the art that some or all ofthe inventive method may be performed using suitably programmed digitalcomputing equipment, or special-purpose hardware.

The foregoing description has been given for a specific illustrativeembodiment of this invention, but the scope of the invention is notlimited to the illustrative embodiment described herein. In general, theinvention will be applicable to similar flow geometries and dynamics asfor the described embodiment. It will be further apparent that variousvariations and modifications may be made to the invention, with theattainment of some or all of the advantages of the invention. It is theobject of the appended claims to cover these and such other variationsand modifications as come within the true spirit and scope of theinvention.

What is claimed is:
 1. A method of detecting the onset of turbulence inconnection with a body moving through a fluid medium comprising thesteps of:supplying in the sidewall of the body, in at least a region ofthe body in which turbulence is expected to occur, sensors each forgenerating a signal suitable for measuring amplitude of pressurefluctuations of the medium proximate a region of said sidewall; during areference stage during which the body moves through the fluid mediumwhen it is known that turbulence is occurring around at least a portionof said body, (i) enabling the sensors to each generate referencetemporal pressure data representing fluctuations in pressure of thefluid medium around said body, and (ii) generating, in response toreference temporal pressure data generated by sensors in a turbulencezone at which turbulence is occurring and sensors in a transition zonebetween the turbulence zone and a laminar flow zone, a method-of-delayphase portrait for each of a progression of selected delay interval,s;and during an operational stage during which the body moves through thefluid medium, (i) enabling the sensors to each generate operationaltemporal pressure data representing fluctuations in pressure of thefluid medium around said body, and generating in response amethod-of-delay phase portrait for each of a progression of selecteddelay intervals, and (ii) comparing the phase portraits, forcorresponding ones of said selected delay intervals, generated inresponse to the operational temporal pressure data during theoperational stage to phase portraits generated in response to thereference temporal pressure data from the transitional zone and theturbulence zone to determine whether the phase portraits generated inresponse to the operational temporal pressure data have selectedcharacteristics present in the phase portraits generated in response tothe reference temporal pressure data from the transitional zone and theturbulence zone.
 2. A method as defined in claim 1 further including thesteps of low-pass filtering said reference temporal pressure data andsaid operational temporal pressure data prior to generating respectivephase portraits.
 3. A method as defined in claim 2 in which low-passfiltering of said reference temporal pressure data is accomplishedaccording to the steps of:performing a Fourier transform operation inconnection with said reference temporal pressure data to generate a setof Fourier coefficients each associated with a frequency; eliminatingfrom said set of Fourier coefficients, those of said Fouriercoefficients associated with frequencies above a predetermined cut-offfrequency; and performing an inverse Fourier transform operation inconnection with remaining Fourier coefficients in said set of Fouriercoefficients.
 4. A method as defined in claim 2 in which low-passfiltering of said operational temporal pressure data is accomplishedaccording to the steps of:performing a Fourier transform operation inconnection with said operational temporal pressure data to generate aset of Fourier coefficients each associated with a frequency;eliminating from said set of Fourier coefficients, those of said Fouriercoefficients associated with frequencies above a predetermined cut-offfrequency; and performing an inverse Fourier transform operation inconnection with remaining Fourier coefficients in said set of Fouriercoefficients.
 5. A method as defined in claim 2 in which low-passfiltering of said reference temporal pressure data and of saidoperational temporal pressure data are both accomplished according tothe steps of:performing Fourier transform operations in connection withsaid reference temporal pressure data and said operational temporalpressure data to generate respective sets of Fourier coefficients eachassociated with a frequency; eliminating from said respective sets ofFourier coefficients, those of said Fourier coefficients associated withfrequencies above a predetermined cut-off frequency; and performinginverse Fourier transform operations in connection with remainingFourier coefficients in said set of Fourier coefficients.
 6. A method asdefined in claim 5 in which said respective sets of Fourier coefficientsare further used to generate power spectra, said turbulence onsetdetermination further being made in response to a comparison of saidpower spectra generated in response to (i) said reference temporalpressure data from sensors in said turbulence zone and sensors in saidtransition zone, and (ii) said operational temporal pressure data.
 7. Asystem for detecting the onset of turbulence in connection with a bodymoving through a fluid medium, the sidewall of the body, in at least aregion of the body in which turbulence is expected to occur, includingsensors each for generating a signal suitable for measuring amplitude ofpressure fluctuations of the medium proximate a region of said sidewall,said system including:first and second means operative during areference stage during which the body moves through the fluid mediumwhen it is known that turbulence is occurring around at least a portionof said body, said first means being operative to enable the sensors toeach generate reference temporal pressure data representing fluctuationsin pressure of the fluid medium around said body, said second meansbeing operative to generate in response to reference temporal pressuredata generated by sensors in a turbulence zone at which turbulence isoccurring and sensors in a transition zone between the turbulence zoneand a laminar flow zone, a method-of-delay phase portrait for each of aprogression of selected delay intervals; third means operative during anoperational stage during which the body moves through the fluid mediumfor enabling the sensors to each generate operational temporal pressuredata representing fluctuations in pressure of the fluid medium aroundsaid body, and generating in response a method-of-delay phase portraitfor each of a progression of selected delay intervals; and fourth meansoperative during the operational stage for comparing the phaseportraits, for corresponding ones of said selected delay intervals,generated in response to the operational temporal pressure data duringthe operational stage to phase portraits generated in response to thereference temporal pressure data from the transitional zone and theturbulence zone to determine whether the phase portraits generated inresponse to the operational temporal pressure data have selectedcharacteristics present in the phase portraits generated in response tothe reference temporal pressure data from the transitional zone and theturbulence zone.